Fock Operator for minimization

[jusy1]

Hi everyone,

I have a question about the application of the Fock operator when we want to optimize, for example, an atomic orbital written as a product of Slater Orbitals.

I know that the appeareance of the Fock operator is due to the minimization of the <E> with respect to each wavefunction, using Lagrange multipliers.

Since we do not make any assumption about the mathematical form of the orbitals we may be using, if my wavefunction had four parameters to optimize:

$$\Psi = c_1(\frac{\zeta_1^3}{\pi})^{1/2}e^{-\zeta_1r} + c_2(\frac{\zeta_2^3}{\pi})^{1/2}e^{-\zeta_2r}$$

Would I get 4 Fock equations, one for each parameter?

Because in most of the derivations of the Fock equation it's written $$dL[{\Psi_i}] = 0$$ But when we are considering an infinitesimal change on our wavefunction it can be on any of its variables: $$\frac{\partial L[{\Psi_i}]}{\partial \Psi_i}\frac{\partial \Psi_i}{\partial r_i}\partial r_i$$

Is this right?

Thank you

 

[eljose79]

for your question. The Fock operator is indeed a key tool in the optimization of atomic orbitals. The Fock operator is derived from the variational principle, which states that the energy of a system will be minimized with respect to the wavefunction. This means that when we apply the Fock operator to a wavefunction, we are essentially finding the best possible approximation to the true wavefunction that will give us the lowest possible energy.

To answer your question, if your wavefunction has four parameters to optimize, then yes, you would need to solve four Fock equations, one for each parameter. This is because each parameter will have a different effect on the overall wavefunction, and therefore will require its own equation to optimize it.

In terms of the infinitesimal change in the wavefunction, you are correct that it can be on any of its variables. This is because the Fock operator takes into account all possible variations in the wavefunction, including changes in its parameters. So, in order to fully optimize the wavefunction, we need to consider all possible changes in its variables.

I hope this helps to clarify the role of the Fock operator in optimizing atomic orbitals. Let me know if you have any other questions.